When \(m=m(s,t)=3\) , the (generalized) star operations defined above recover the original star operations introduced
by Kazhdan and Lusztig in [1]}. In this case at most one of \(w^*\)  and \(w_*\)  is defined depending on the value
of \(w_I\) , therefore we may unambiguously speak of the right star operation with respect to \(I\), without
indicating if the operation is upper or lower. More precisely, both \(w^*\)  and \(w_*\)  are undefined if \(w_I\in \lbrace 1,sts=tst\rbrace \) , \(w^*\)  is defined while \(w_*\)  is not if \(w_I\in \lbrace s,t\rbrace \) , and \(w_*\)  is defined while \(w^*\)  is not if
\(w_I\in \lbrace st,ts\rbrace \) . In the last two cases, we denote the element among \(w^*\)  and \(w_*\)  that is defined by \(w*\) ,
placing the star sign in the middle. We call the operation \(w\mapsto w*\)  a simple star operation; it is an
involution in the sense that \((w*)*=w\)  whenever \(w*\)  is defined. Similarly, the pair \(I=\lbrace s,t\rbrace \)  gives rise to a
partially defined involution \(w\mapsto *w\)  on \(W\)  that we call the simple left star operation with respect to
\(I\).
